Imagine I have unbounded $d$-dimensional integer lattice where I take two vertices, $v_a$ and $v_b$, separated by a fixed Manhattan distance $L$, and I release a random walker at $v_a$ and allow for absorption (with a probability of unity) at $v_b$. How does the probability of absorption and the mean first passage time (MFPT) for absorption at $v_b$ scale with $L$?

Polya demonstrated the the origin recurrence probability, $p(d)$, of a random walker on a $d$-dimensional integer lattice is unity for $d = {1,2}$ and that:

$p(3) = \frac{6^{\frac{1}{2}}}{32*\pi^3} * \Gamma(\frac{1}{24}) * \Gamma(\frac{5}{24}) * \Gamma(\frac{7}{24}) * \Gamma(\frac{11}{24})$

( http://mathworld.wolfram.com/PolyasRandomWalkConstants.html )

From Polya's result I would guess that if $L \approx 1$, the probability of absorption at $v_b$ would be $\approx p(3)$. However, that's simply a guess, and offers little information on the MFPT for absorption.

What might change if we instead consider a Brownian motion?

Update :: I am most interested in a good estimate for how the absorption probability and MFPT scales as $L$ goes from $1$ to $\infty$, rather than an asymptotic.

Update 2 :: I have written a post on mathematics stackexchange asking for further explanation of Omer's answer. My concern was that such a discussion might be too low level for this forum. I hope this is an appropriate thing to do.

Update 3 :: I'm simulated random walks on an infinite $Z^3$ integer lattice, where $10^5$ steps without absorbence at a target vertex (near the origin) counts as the walker diverging to infinity. Walks are initialized at the origin, (0,0,0), and values for means-square-displacement (MSD) and the number of steps prior to absorption are averages over $10^3$ iterations.

Absorbing target = {0,0,0}

Fraction of absorbed walks prior to 10^5 steps = 353/1000 = 35.3%

Mean displacement of walker = 279.824

Mean[# steps until absorbance or 10^5 steps] = 64731.3

Mean[# steps conditioned on absorbance] = 88.7

Absorbing target = {0,0,1}

Fraction of absorbed walks prior to 10^5 steps = 335/1000 = 33.5%

Mean displacement of walker = 288.447

Mean[# steps until absorbance or 10^5 steps] = 66628.2

Mean[# steps conditioned on absorbance] = 382.7

Absorbing target = {0,0,2}

Fraction of absorbed walks prior to 10^5 steps = 155/1000 = 15.5%

Mean displacement of walker = 367.702

Mean[# steps until absorbance or 10^5 steps] = 84556.8

Mean[# steps conditioned on absorbance] = 366.5

Absorbing target = {0,0,3}

Fraction of absorbed walks prior to 10^5 steps = 114 / 1000 = 11.4%

Mean displacement of walker = 385.576

Mean[# steps until absorbance or 10^5 steps] = 88642.4

Mean[# steps conditioned on absorbance] = 371.9

Absorbing target = {0,0,15}

Fraction of absorbed walks prior to 10^5 steps = 16 / 1000 = 1.6%

Mean displacement of walker = 430.08

Mean[# steps until absorbance or 10^5 steps] = 98427.1

Mean[# steps conditioned on absorbance] = 1693.8

Absorbing target = {0,0,30}

Fraction of absorbed walks prior to 10^5 steps = 9 / 1000 = 0.9%

Mean displacement of walker = 440.352

Mean[# steps until absorbance or 10^5 steps] = 99161.4

Mean[# steps conditioned on absorbance] = 6822.2

1more comment